Knowing how costs change as output changes is essential to: A. planning and controlling. B. controlling and decision making. C. planning, controlling and decision making. D. decision making and planning. (4pts) Question 11 - Knowing how costs change as output changes is essential to: A. planning and controlling. B. controlling and decision making. C. planning, controlling and decision making. D. decision making and planning.
3. (a) During lunch hour, arrivals of customers at a pizza hut restaurant follows a Poisson process with the rate of 120 customers per hour. The restaurant has one line, with three workers taking food orders at independent services stations. Each worker takes an exponen- tially distributed amount of time-on average 1 minute-to serve a customer. Let X, denote the number of customers in the restaurant (in line and being serviced) at time t. Then, the process (Xt : 1 2 0) is a continuous-time Markov chain. (i) Show that the process is a birth-and-death process by giving the birth and death rates. [4 marks] (ii) Find the generator matrix of the above process. [5 marks] (iii) For each integer & 2 0, derive the long-term probability that there are & customers in the restaurant. [8 marks] (iv) Calculate the long-term probability that all three workers are busy. [6 marks] (v) Find the average number of customers in the restaurant in the long term. [7 marks ] (b) A student support center has 3 tutors who help students with their home work. Students arrive at the center according to a Poisson process at rate A = 3 per hour. Each tutor's service time is exponentially distributed with average of 1/10 hours. Tutors' service times and student arrival times are independent. If all the tutors are busy when a student arrives at the center, the student will leave. Let X, denote the number of tutors who are busy at time t. Determine its generator matrix and stationary distribution. [10 marks]PRoBLEM 10.4 A chemical solution contains N molecules of type A and an equal number of molecules of type B. A reversible reaction occurs between type A and B molecules in which they bond to form a new compound AB. Suppose that in any small time interval of length h, any particular unbonded A molecule will react with any particular unbonded B molecule with probability och + O[h), where or is a reaction rate of formation. Suppose also that in any small time interval of length hI any particular AB molecule disassociates into its A and B constituents with probability ll + DUI), where fl is a reaction rate of dissolution. Let X (t) denote the number of AB molecules at time t. Model X(t} as a birth and death process by specifying the parameters. Note that one mole of molecules is N = 6.02214129 X 1023. Problem 2: 10 points Consider a birth-and-death process, X = {X(t) : t 2 0} , associated with the service line that consists of N = 10 servers. When all 10 servers are occupied, the new request is refused and not coming into the service line. As there are k k =3. (10 -k) for 0 10, and P [X(t + 4) - X(t) = -1\\X(t) = k] =/k = 2 . k for 0
